Matrix Wigner Function and SU(1,1)

Abstract: This paper contains a brief sketch of some methods that can be used to obtain the Wigner function for a number of systems. We give an overview of the technique as it is applied to some simple differential systems related to diffusion problems in one dimension. We compute the Wigner function for the harmonic oscillator, the $xp$ interaction, and a hyperbolic oscillator. These systems are shown to share several properties in common related to the Whittaker function and various formulae for the Laguerre polynomials. To contrast with the techniques that are applicable to problems involving continuous states, we then show that by expanding the solution space to the hyperbolic plane and utilising some results from matrix calculus, we are able to recover a number of interesting identities for SU(1,1) and the pseudosphere. We close with a discussion of some more advanced topics in the theory of the Wigner function.